\titleofpresentation{Characters and Quasi-Permutation Representations of
$2$-Groups with $\left|\Omega_2(G)\right|=16$}

\presenter{Ghodrat Ghaffarzadeh}

By a quasi-permutation matrix we mean a square matrix over the
complex field $\mathbb C$ with non-negative integral trace. Thus
every permutation matrix over $\mathbb C$ is a quasi-permutation
matrix. For a given finite group $G$, let $p(G)$ denote the
minimal degree of a faithful permutation representation of $G$ (or
of a faithful representation of $G$ by permutation matrices), let
$q(G)$ denote the minimal degree of a faithful representation of
$G$ by quasi-permutation matrices over the rational field $\mathbb
Q$, and let $c(G)$ denote the minimal degree of a faithful
representation of $G$ by complex quasi-permutation matrices. It is
easy to see that
\[c(G) \leq q(G) \leq p(G) \]
where $G$ is a finite group.  In \cite{ghaffarzadeh3}, finite $2-$groups $G$
have been determined with the property $\mid\Omega_2(G)\mid=16$.
We recall that ${\Omega}_2(G)=<x\in G : x^4=1>$. In this paper,
we study characters and quasi-permutation representations of this
groups which are cyclic center and have the additional property
that $G$ is non-abelian and has a normal elementary abelian
subgroup $E$ of order $8$. In \cite{ghaffarzadeh3}, these groups have been
determined in terms of generators and relations. Also, it is
shown that $G/E$ is either cyclic or a generalized quaternion
$Q_{2^n}$ of order $2^n$, $n\geq 3$. If $G/E$ is cyclic, then
there
 exists one class of $2-$groups \cite[theorem 3.1]{ghaffarzadeh3}, and if
 $G/E$ is generalized quaternion group, then there exist two
 classes of $2-$groups \cite[theorems 3.3 and 3.4]{ghaffarzadeh3}.
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\begin{ourbiblio}{99}
\bibitem{ghaffarzadeh1}
{H. BEHRAVESH}. {\em Quasi-permutation  representations of
$p$-groups of class $2$}. {\rm  J.London Math. Soc. (2) 55 (1997)
251-260}.
\bibitem{ghaffarzadeh2}
{C. W. CURTIS AND I. REINER}. {\em Representation theory of
finite groups and associative algebras}. {\rm Wiley-Interscience,
New York, 1962}.
 \bibitem{ghaffarzadeh3}
{ZVONIMIR JANKO}. {\em Finite $2-$Groups $G$ with
$\Omega_2(G)=16$}. {\rm Glasnic J. Math., 2004}.
 \end{ourbiblio}
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\presenteraddress{University of Urmia, Urmia, Iran}

\email{q_gafarzadeh@yahoo.com}

%\web{}
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